8.4 Transformers

Basics of Transformers

A transformer transfers electrical energy between two circuits using electromagnetic induction. It allows you to change voltage levels without a direct electrical connection between the input and output sides. This is what makes long-distance power transmission practical: by stepping voltage up, you reduce current and therefore reduce energy lost as heat in the wires.

Transformers tie together several core Physics II concepts, especially Faraday's law and mutual inductance. If you understand how a changing magnetic field induces an EMF, you already understand the basic principle behind every transformer.

Definition and Purpose

A transformer consists of two coils of wire (called windings) wrapped around a shared magnetic core. An AC voltage applied to one winding creates a changing magnetic flux in the core, which induces a voltage in the other winding.

Key functions:

• Voltage transformation: step voltage up or down depending on the application
• Efficient power transmission: higher voltage means lower current, which means less $I^2R$ loss in transmission lines
• Galvanic isolation: the primary and secondary circuits have no direct electrical connection, which provides safety and noise reduction

Components of a Transformer

• Primary winding: the coil connected to the AC power source
• Secondary winding: the coil that delivers power to the load
• Magnetic core: usually made of laminated iron or steel, this concentrates the magnetic flux so nearly all of it links both windings
• Insulation: prevents short circuits between turns and between windings
• Cooling system: oil-filled or air-cooled designs manage the heat produced during operation

Types of Transformers

Different applications call for different transformer designs:

• Power transformers handle high voltages in the electrical grid (transmission level)
• Distribution transformers step voltage down for neighborhoods and buildings
• Instrument transformers let you safely measure high voltages and currents
• Audio transformers match impedances in sound systems
• Pulse transformers transmit digital signals while maintaining electrical isolation

Electromagnetic Principles

Transformer operation rests on a few electromagnetic laws you've already studied. Here's how each one connects to what a transformer actually does.

Faraday's Law of Induction

Faraday's law says that a changing magnetic flux through a coil induces an EMF (voltage) in that coil. The faster the flux changes, the larger the induced EMF.

$\varepsilon = -N\frac{d\Phi}{dt}$

where $N$ is the number of turns and $\Phi$ is the magnetic flux through each turn. The negative sign reflects Lenz's law: the induced EMF opposes the change that created it.

In a transformer, AC current in the primary winding creates a continuously changing magnetic flux in the core. That changing flux passes through the secondary winding and induces a voltage there. This is the entire mechanism of energy transfer.

Mutual Inductance

Mutual inductance $M$ quantifies how effectively a changing current in one coil induces a voltage in a nearby coil. It depends on the geometry of both coils, the number of turns, and the core material.

$M = \frac{N_1 N_2}{R_m}$

Here $R_m$ is the magnetic reluctance of the core (think of it as the core's "resistance" to magnetic flux). A low-reluctance core, like one made of iron, gives high mutual inductance and better energy transfer. Air gaps in the core increase reluctance and reduce coupling.

Magnetic Flux Linkage

Flux linkage $\lambda$ is the total flux "seen" by a coil, accounting for every turn:

$\lambda = N\Phi$

This quantity directly determines the induced voltage in each winding. A core with high permeability and large cross-sectional area maximizes flux linkage, which is why transformer cores are made from materials that conduct magnetic flux well.

Transformer Construction

How a transformer is built has a direct impact on its efficiency, size, and cost. The main design decisions involve the core, the windings, and the insulation.

Core Materials

• Laminated silicon steel is the standard for power-frequency (50/60 Hz) transformers. The thin laminations reduce eddy current losses because they limit the paths available for induced currents in the core.
• Ferrite cores work well at high frequencies (kHz to MHz range) where steel would have excessive losses.
• Amorphous metal cores have very low hysteresis losses and are used in high-efficiency distribution transformers.
• Air cores are used at very high frequencies where any magnetic material would introduce too much loss.
• Core geometry matters too: shell-type cores surround the windings with the core, while core-type designs wrap the windings around the core legs. Each affects the flux path differently.

Winding Techniques

• Layer winding: turns are wound in neat layers with insulation between them; good for general-purpose transformers
• Disc winding: used in high-voltage transformers to reduce leakage inductance
• Foil winding: thin metal sheets replace round wire in low-voltage, high-current designs to minimize skin effect
• Interleaved windings: alternating sections of primary and secondary improve magnetic coupling
• Tap windings: extra connection points along a winding allow voltage adjustment

Insulation Methods

• Oil immersion provides both electrical insulation and cooling in large power transformers
• Dry-type transformers rely on air and solid insulating materials, common in indoor installations
• Epoxy encapsulation protects against moisture and contaminants
• Paper-oil systems combine cellulose paper with mineral oil in large units
• Enamel coating on magnet wire provides turn-to-turn insulation in smaller transformers

Transformer Operation

Primary vs. Secondary Coils

When you apply an AC voltage to the primary coil, it draws current that creates a changing magnetic flux in the core. That flux threads through the secondary coil and induces a voltage there.

The number of turns in each coil determines the voltage ratio. The primary current creates a magnetomotive force (MMF) in the core. When the secondary is connected to a load and current flows, it creates its own opposing MMF. The primary then draws additional current to maintain the core flux, which is how power transfers from source to load.

Step-Up vs. Step-Down Transformers

• A step-up transformer has more turns on the secondary than the primary, so the output voltage is higher than the input voltage. Power plants use these to boost voltage to hundreds of kilovolts for long-distance transmission.
• A step-down transformer has fewer turns on the secondary, producing a lower output voltage. Substations and the cylindrical transformers on utility poles are step-down transformers that bring voltage to usable levels (like 120 V or 240 V for homes).

The key trade-off: when voltage goes up, current goes down (and vice versa), because power is conserved.

Turns Ratio

The turns ratio is the single most important design parameter:

$n = \frac{N_s}{N_p}$

It directly gives you the voltage relationship:

$\frac{V_s}{V_p} = \frac{N_s}{N_p}$

And the current relationship is the inverse:

$\frac{I_s}{I_p} = \frac{N_p}{N_s}$

Example: A transformer with 500 turns on the primary and 50 turns on the secondary has a turns ratio of $n = 50/500 = 0.1$. If you apply 120 V to the primary, the secondary voltage is $120 \times 0.1 = 12$ V. If the primary current is 0.5 A, the secondary current is $0.5 / 0.1 = 5$ A.

Ideal Transformer Theory

The ideal transformer model assumes no energy losses of any kind. While no real transformer is perfect, this model captures the essential behavior and is the starting point for all transformer calculations.

Assumptions of an ideal transformer:

• Zero winding resistance (no copper losses)
• Perfect magnetic coupling (all flux links both windings)
• Infinite core permeability (no magnetizing current needed)
• No core losses (no hysteresis or eddy currents)

Voltage and Current Relationships

For an ideal transformer:

$\frac{V_s}{V_p} = \frac{N_s}{N_p}$

$\frac{I_p}{I_s} = \frac{N_s}{N_p}$

These two equations let you find any unknown voltage or current if you know the turns ratio and one other quantity.

Power Conservation

Since an ideal transformer has no losses, power in equals power out:

$P_p = P_s$

$V_p I_p = V_s I_s$

This is why stepping up voltage necessarily means stepping down current. The product $V \times I$ stays constant. This relationship is extremely useful for quick calculations and serves as a good approximation even for real transformers, which typically operate at 95%+ efficiency.

Impedance Transformation

A transformer doesn't just transform voltage and current. It also transforms impedance. If a load impedance $Z_s$ is connected to the secondary, the impedance "seen" by the source on the primary side is:

$Z_p = \left(\frac{N_p}{N_s}\right)^2 Z_s = \frac{Z_s}{n^2}$

This is called reflected impedance. It's the reason transformers are used for impedance matching: you can make a source "see" whatever impedance it needs for maximum power transfer by choosing the right turns ratio. This comes up in audio systems, RF circuits, and antenna feed networks.

Real Transformer Considerations

Real transformers lose some energy during operation. Understanding where these losses come from helps you predict efficiency and choose the right transformer for a given application.

Core Losses

Core losses occur in the magnetic core itself and are present even with no load connected:

• Hysteresis losses: energy spent re-magnetizing the core every AC cycle. The core material's B-H curve determines how much energy is lost per cycle. Materials with narrow hysteresis loops (like silicon steel) have lower losses.
• Eddy current losses: the changing flux induces small circulating currents in the core material itself. These currents dissipate energy as heat. Laminating the core into thin, insulated sheets greatly reduces eddy currents by breaking up the current paths.

Both types of loss increase with frequency and with higher flux density in the core.

Copper Losses

Copper losses (also called $I^2R$ losses or winding losses) come from the electrical resistance of the wire in both windings. They're proportional to the square of the current, so they increase significantly under heavy load.

To reduce copper losses, designers use thicker wire (lower resistance) or parallel conductors. These losses are the dominant source of inefficiency at high loads.

Efficiency and Power Factor

Transformer efficiency is:

$\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{core} + P_{copper}}$

Large power transformers achieve efficiencies of 95% to 99%. Smaller transformers tend to be less efficient.

The power factor at no load can be quite low because the transformer draws a magnetizing current that is mostly reactive (out of phase with the voltage). Under load, the power factor improves as the resistive component of current dominates.

Transformer Applications

Power Distribution Systems

The electrical grid depends on transformers at every stage:

1. Generation: a power plant produces electricity at moderate voltage (e.g., 11-25 kV)
2. Transmission: step-up transformers boost this to very high voltage (115-765 kV) for long-distance lines, minimizing $I^2R$ losses
3. Substation: step-down transformers reduce voltage to distribution levels (4-35 kV)
4. Final delivery: pole-mounted or pad-mounted transformers step voltage down to 120/240 V for homes or 480 V for commercial buildings

Without transformers, you'd need impossibly thick cables to transmit power over long distances at low voltage.

Electronic Circuits

Transformers appear in many electronic applications:

• Isolation transformers protect users and equipment by eliminating direct electrical connections (critical in medical devices)
• Pulse transformers pass digital signals while maintaining galvanic isolation
• Audio transformers match impedances between amplifiers and speakers
• Flyback transformers generate high voltages in switch-mode power supplies
• Current transformers allow safe measurement of large currents by stepping them down to manageable levels

Isolation Transformers

These have a 1:1 turns ratio, so they don't change the voltage. Their purpose is purely to provide galvanic isolation between circuits. This breaks ground loops, reduces electrical noise, and prevents shock hazards. You'll find them in medical equipment, laboratory instruments, and data communication systems where safety and signal integrity are critical.

Transformer Testing

Three standard tests characterize a transformer's real-world behavior and provide the parameters needed for its equivalent circuit model.

Open-Circuit Test

Purpose: Measure core losses and magnetizing current.

1. Leave the secondary winding open (no load connected)
2. Apply rated voltage to the primary at rated frequency
3. Measure the primary voltage, current, and power

Since no current flows in the secondary, the small current drawn by the primary is the magnetizing current. The power measured is almost entirely core loss (hysteresis + eddy currents). This test gives you the shunt branch parameters of the equivalent circuit.

Short-Circuit Test

Purpose: Measure winding resistance and leakage reactance.

1. Short-circuit the secondary winding
2. Apply a reduced voltage to the primary, just enough to produce rated current
3. Measure the primary voltage, current, and power

The applied voltage is typically only 5-10% of rated voltage. At this low flux level, core losses are negligible, so the measured power represents copper losses. This test gives you the series branch parameters of the equivalent circuit.

Load Test

Purpose: Evaluate overall performance under realistic conditions.

1. Apply rated voltage to the primary
2. Connect a variable load to the secondary
3. Measure efficiency, voltage regulation, and temperature rise at various load levels

This test confirms that the transformer meets its performance specifications and can handle its rated load without overheating.

Special Transformer Types

Autotransformers

An autotransformer uses a single winding with a tap, so part of the winding is shared between the input and output. This makes it smaller, lighter, and more efficient than a two-winding transformer of the same rating. The trade-off is that there's no galvanic isolation between primary and secondary.

Autotransformers are commonly used in variable voltage supplies (like a Variac), motor starters, and situations where the voltage change is relatively small.

Three-Phase Transformers

Industrial power systems use three-phase AC, and three-phase transformers handle all three phases in a single unit. The windings can be connected in delta (triangle) or wye (star) configurations on either side, and the specific combination affects the phase relationship between input and output (called the vector group).

A single three-phase transformer is more compact and efficient than using three separate single-phase transformers.

Instrument Transformers

These are precision devices used for measurement and protection:

• Current transformers (CTs) step down large currents to small, measurable values (e.g., 1000 A down to 5 A). The secondary of a CT should never be left open while the primary carries current, as dangerously high voltages can develop.
• Potential transformers (PTs) step down high voltages to safe levels for meters and protective relays (e.g., 13.8 kV down to 120 V).

Both types provide isolation between the high-voltage circuit and the measuring instruments.

Transformer Safety

Overheating Prevention

Transformers generate heat from both core and copper losses. If temperature rises too high, insulation breaks down and the transformer can fail catastrophically.

• Temperature sensors and monitoring systems detect abnormal heating
• Cooling systems (oil pumps, radiators, fans) keep temperatures within safe limits
• Thermal overload protection automatically disconnects the transformer if it overheats
• Proper ventilation around the transformer enclosure prevents heat buildup
• Regular maintenance of cooling systems is essential for long-term reliability

Electrical Isolation

• Insulation systems rated for the transformer's voltage class prevent contact with live parts
• Bushings provide insulated pathways for conductors entering and exiting the transformer tank
• Protective enclosures and barriers restrict physical access to high-voltage components
• Grounding the transformer tank and secondary neutral provides a safe fault path

Grounding Practices

Proper grounding is critical for transformer safety:

• The transformer tank (metal enclosure) must be grounded to prevent shock if insulation fails
• Neutral grounding on the secondary side limits overvoltages during faults
• Grounding resistors or reactors can be used to limit fault current magnitude in large systems
• Ground connections should be tested regularly to verify they remain effective
• All grounding must comply with applicable electrical codes (such as the NEC)